Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
− whenever |y1 − z1| > δ then θ = θ1, and the delay timer is reset,
2.6 On Adaptive High-gain Observers
− when |y1 − z1| < δ, we don’t know wether it is an overshoot or not: θ = θ1 and the
delay timer is started,
− when |y1 − z1| < δ and the delay timer is equal to Td, estimation is satisfactory and
θ = θ2.
The authors consider a control strategy that stabilizes the system provided that the
full state is known. They include a fixed high-gain observer and consider the closed loop
system. They propose a set of assumptions such that the system-observer-controler ensemble
is uniformly asymptotically stable 20 , (see [11], Theorem 1). The last step is the demonstration
that stability remains when the high-gain of the observer is switched between two well defined
values (see [11], Theorem 2 and example of Section 4).
The Luemberger observer doesn’t have the same local properties as the extended Kalman
filter, namely good filtering properties and analytically guaranteed convergence for small
initial errors. We therefore expect a high-gain extended Kalman filter having a varying θ
parameter to be more efficient with respect to the noise filtering issue.
y z
2.6.4 Adaptive Kalman Filters
Overshoots=no high gain change
Td
Change in the high gain value
Time
Figure 2.2: Switching strategy to deal with peaking.
We now consider the problem of the adaptation of the high-gain parameter of Kalman
filters. Recall that the high-gain parameter is used to provide the Q and R matrices with a
specific structure. Therefore, the adaptation of the high-gain parameter may be seen as the
modification of those two matrices 21 . There is nonetheless a big difference when the high-gain
structure is not considered: there is no proof of convergence of the extended Kalman filter
when the estimated state doesn’t lie in a neighborhood of the real state. The situation could
be even worse. Examples of systems for which the filter doesn’t converge can be found in
[100, 101].
The adaptation problem, when viewed as the adaptation of the Q and R matrices, has
been the object of quite a few publications both in linear and nonlinear cases. In the linear
20 The Theorem demonstrates that all the trajectories are bounded (ultimately with time) and that the
trajectory (under output feedback i.e. using the observer) is close to that of the state feedback (i.e. controller
with full state knowledge).
21 Recall that when the system is modeled using stochastic differential equations, Q and R represent the
covariances of the state and output noise, respectively.
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